In this paper, we consider a class of orientation-preserving Morse–Smale diffeomorphisms defined on an orientable surface. The papers by Bezdenezhnykh and Grines showed that such diffeomorphisms have a finite number of heteroclinic orbits. In addition, the classification problem for such diffeomorphisms is reduced to the problem of distinguishing orientable graphs with substitutions describing the geometry of a heteroclinic intersection. However, such graphs generally do not admit polynomial discriminating algorithms. This article proposes a new approach to the classification of these cascades. For this, each diffeomorphism under consideration is associated with a graph that allows the construction of an effective algorithm for determining whether graphs are isomorphic. We also identified a class of admissible graphs, each isomorphism class of which can be realized by a diffeomorphism of a surface with an orientable heteroclinic. The results obtained are directly related to the realization problem of homotopy classes of homeomorphisms on closed orientable surfaces. In particular, they give an approach to constructing a representative in each homotopy class of homeomorphisms of algebraically finite type according to the Nielsen classification, which is an open problem today.
In this paper, we consider regular topological flows on closed n-manifolds. Such flows have a hyperbolic (in the topological sense) chain recurrent set consisting of a finite number of fixed points and periodic orbits. The class of such flows includes, for example, Morse – Smale flows, which are closely related to the topology of the supporting manifold. This connection is provided by the existence of the Morse –Bott energy function for the Morse – Smale flows. It is well known that, starting from dimension 4, there exist nonsmoothing topological manifolds, on which dynamical systems can be considered only in a continuous category. The existence of continuous analogs of regular flows on any topological manifolds is an open question, as is the existence of energy functions for such flows. In this paper, we study the dynamics of regular topological flows, investigate the topology of the embedding and the asymptotic behavior of invariant manifolds of fixed points and periodic orbits. The main result is the construction of the Morse –Bott energy function for such flows, which ensures their close connection with the topology of the ambient manifold.
This paper is a continuation of research in the direction of energy function (a smooth Lyapunov function whose set of critical points coincides with the chain recurrent set of a system) construction for discrete dynamical systems. The authors established the existence of an energy function for any AA-diffeomorphism of a three-dimensional closed orientable manifold whose non-wandering set consists of a chaotic one-dimensional canonically embedded surface attractor and repeller.
We study the geometry of the bifurcation diagrams of the families of vector fields in the plane. Countable number of pairwise non-equivalent germs of bifurcation diagrams in the two-parameter families is constructed. Previously, this effect was discovered for three parameters only. Our example is related to so-called saddle node (SN)–SN families: unfoldings of vector fields with one saddle-node singular point and one saddle-node cycle. We prove structural stability of this family. By the way, the tools that may be helpful in the proof of structural stability of other generic two-parameter families are developed. One of these tools is the embedding theorem for saddle-node families depending on the parameter. It is proved at the end of the paper.
The role of various long-wave approximations in the description of the wave field and bottom pressure caused by surface waves, and their relation to evolution equations are being considered. In the framework of the linear theory, these approximations are being tested on the well-known exact solution for the wave spectral amplitudes and pressure variations. The famous Whitham, Korteweg–de Vries (KdV) and Benjamin–Bona–Mahony (BBM) equations have been used as evolutionary equations. It has been shown that if the wave is long, though steep enough, the BBM approximation gives better results than the KdV approximation, and they are quite close to the exact results. The same applies to the description of rogue waves, though formed from smooth relatively long waves, are often short and steep, then they may be invisible in variations of the bottom pressure. Another advantage of the BBM approximation for calculating the bottom pressure is the ability to analyze noisy series without preliminary filtering, which is necessary when using the KdV approximation.
Meteotsunamis are long waves generated by displacement of a water body due to atmospheric pressure disturbances that have similar spatial and temporal characteristics to landslide tsunamis. NAMI DANCE that solves the nonlinear shallow water equations is a widely used numerical model to simulate tsunami waves generated by seismic origin. Several validation studies showed that it is highly capable of representing the generation, propagation and nearshore amplification processes of tsunami waves, including inundation at complex topography and basin resonance. The new module of NAMI DANCE that uses the atmospheric pressure and wind forcing as the other inputs to simulate meteotsunami events is developed. In this paper, the analytical solution for the generation of ocean waves due to the propagating atmospheric pressure disturbance is obtained. The new version of the code called NAMI DANCE SUITE is validated by comparing its results with those from analytical solutions on the flat bathymetry. It is also shown that the governing equations for long wave generation by atmospheric pressure disturbances in narrow bays and channels can be written similar to the 1D case studied for tsunami generation and how it is integrated into the numerical model. The analytical solution of the linear shallow water model is defined, and results are compared with numerical solutions. A rectangular shaped flat bathymetry is used as the test domain to model the generation and propagation of ocean waves and the development of Proudman resonance due to moving atmospheric pressure disturbances. The simulation results with different ratios of pressure speed to ocean wave speed (Froude numbers) considering sub-critical, critical and super-critical conditions are presented. Fairly well agreements between analytical solutions and numerical solutions are obtained. Additionally, basins with triangular (lateral) and stepwise shelf (longitudinal) cross sections on different slopes are tested. The amplitudes of generated waves at different time steps in each simulation are presented with discussions considering the channel characteristics. These simulations present the capability of NAMI DANCE SUITE to model the effects of bathymetric conditions such as shelf slope and local bathymetry on wave amplification due to moving atmospheric pressure disturbances.
On December 22, 2018, a destructive tsunami related to the phenomena caused by the volcanic eruption of Gunung Anak Krakatau (GAK) was generated following a partial collapse of the volcano that caused serious damage and killed more than 400 people. This recent event challenged the traditional understanding of tsunami hazard, warning and response mechanisms and raised the topic of volcanic tsunami hazard. The complex mechanism of this tsunamigenic volcano collapse still needs further investigation as Anak Krakatau is one of the potentially tsunamigenic volcanoes in the world. This study investigates the possible source mechanisms of this phenomenon and their contribution to explaining the observed sea level disturbances by considering the impacts of this destructive event. We configured a flank collapse scenario with a volume of 0.25 km3 as a combination of submarine and subaerial mass movement as the possible source scenarios to the December 22, 2018 Sunda Strait tsunami. A two-layer model is applied to simulate the tsunami generation by these landslides up to 420 s. The tsunami propagation and inundation are then simulated by NAMI DANCE model in GPU environment. The simulation results suggest that this scenario seems capable of generating such a tsunami observed along the coast of Sunda Strait. However, the contribution of the possible submarine mass movements in the close area between GAK and the surrounding islands either to this event or potential tsunami threat in the region is still questionable. We employed a bathymetric dataset through pre- and post-event analyses, which demonstrate submarine slope failures in the southwestern proximity of GAK. Hence, additional two scenarios of elliptical landslide sources on the slopes of bathymetry change area (could be triggered by seismic motion/volcanic eruption) are considered, searching for the possible effects of the tsunami that might be generated by these submarine landslides. The study may also provide some perspective for potential tsunami generation by combined sources and help to elucidate the extent of volcanic tsunami hazard in the region due to potential future eruptions of Gunung Anak Krakatau.
Bifurcations that occur in a small neighborhood of a polycycle of a planar vector field are called semilocal. We prove that even semilocal bifurcations of hyperbolic polycycles may have numeric invariants of topological classification
The existence problem for attractors of foliations with transverse linear connection is investigated. In general foliations with transverse linear connection do not admit attractors. A conditions that implies the existence of a global attractor which is a minimal set, is specified. An application to transversely similar pseudo-Riemannian foliations is obtained. The global structure of transversely similar Riemannian foliations is described. Different examples are constructed.
It is a well known fact that any smooth manifold admits a Morse function, whereas the problem of existence of a Morse function for a topological manifold stated by Marston Morse in 1959 is still open. In the present paper we prove that a topological manifold admits a continuous Morse function if it admits a topological flow with a finite hyperbolic chain recurrent set. We construct this function as a Lyapunov function whose set of the critical points coincides with the chain recurrent set of the flow.
We consider a class of gradient-like flows on three-dimensional closedmanifolds whose attractors and repellers belongs to a finite union of embedded surfaces and find conditions when the ambient manifold is Seifert.
The article is devoted to interrelations between an existence of trivial and nontrivial basic sets of A-diffeomorphisms of surfaces. We prove that if all trivial basic sets of a structurally stable diffeomorphism of surface $M^2$ are source periodic points $\alpha_1, …, \alpha_k$, then the non-wandering set of this diffeomorphism consists of points $\alpha_1, …, \alpha_k$ and exactly one one-dimensional attractor $\Lambda$. We give some sufficient conditions for attractor $\Lambda$ to be widely situated. Also, we prove that if a non-wandering set of a structurally stable diffeomorphism contains a nontrivial zero-dimensional basic set, then it also contains source and sink periodic points.
The observation of a wave group persisting for more than 200 periods in the direct numerical simulation of nonlinear unidirectional irregular water waves in deep water is discussed. The simulation conditions are characterized by parameters realistic for broad-banded waves in the sea. Through solution of the associated scattering problem for the nonlinear Schr€odinger equation, the group is identified as the intense envelope soliton with remarkably stable parameters. Most of the extreme waves occur on top of this group, resulting in higher and longer rogue wave events.
For area-preserving H'enon-like maps and their compositions, we consider smooth perturbations that keep the reversibility of the initial maps but destroy their conservativity. For constructing such perturbations, we use two methods, a new method based on reversible properties of maps written in the so-called cross-form, and the classical Quispel-Roberts method based on a variation of involutions of the initial map. We study symmetry breaking bifurcations of symmetric periodic orbits in reversible families containing quadratic conservative orientable and nonorientable H'enon maps as well as a product of two H'enon maps whose Jacobians are mutually inverse.
An analytic reversible Hamiltonian system with two degrees of freedom is studied in a neighborhood of its symmetric heteroclinic connection made up of a symmetric saddle-center, a symmetric orientable saddle periodic orbit lying in the same level of a Hamiltonian, and two non-symmetric heteroclinic orbits permuted by the involution. This is a codimension one structure; therefore, it can be met generally in oneparameter families of reversible Hamiltonian systems. There exist two possible types of such connections depending on how the involution acts near the equilibrium. We prove a series of theorems that show a chaotic behavior of the system and those in its unfoldings, in particular, the existence of countable sets of transverse homoclinic orbits to the saddle periodic orbit in the critical level, transverse heteroclinic connections involving a pair of saddle periodic orbits, families of elliptic periodic orbits, homoclinic tangencies, families of homoclinic orbits to saddle-centers in the unfolding, etc. As a by-product, we get a criterion of the existence of homoclinic orbits to a saddle-center.
Localized stationary solutions of nonlinear nonlocal Whitham equation with resonance dispersion relation are considered. The existence of exponentially localized smooth and singular solitons, bound states of the solitons and localized solutions with oscillating asymptotics is recognized. The velocity spectra of solitons, in contrast to all other known Whitham equations, appear to be discrete. Asymptotic “quantization rules” for calculation of discrete spectra are obtained.
In the present paper, a solution to the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere is obtained. It is precisely shown that with respect to the stable isotopic connectedness relation there exists countable many of equivalence classes of such systems.
Compactons are studied in the framework of the Korteweg–de Vries (KdV) equation with the sublinear nonlinearity. Compactons represent localized bell-shaped waves of either polarity which propagate to the same direction as waves of the linear KdV equation. Their amplitude and width are inverse proportional to their speed. The energetic stability of compactons with respect to symmetric compact perturbations with the same support is proven analytically. Dynamics of compactons is studied numerically, including evolution of pulse-like disturbances and interactions of compactons of the same or opposite polarities. Compactons interact inelastically, though almost restore their shapes after collisions. Compactons play a two-fold role of the long-living soliton-like structures and of the small-scale waves which spread the wave energy.
The problem of the existence of an arc with at most countable (finite) number of bifurcations connecting structurally stable systems (Morse – Smale systems) on manifolds was included in the list of fifty Palis – Pugh problems at number 33. In 1976 S. Newhouse, J.Palis, F.Takens introduced the concept of a stable arc connecting two structurally stable systems on a manifold. Such an arc does not change its quality properties with small changes. In the same year, S.Newhouse and M.Peixoto proved the existence of a simple arc (containing only elementary bifurcations) between any two Morse – Smale flows. From the result of the work of J. Fliteas it follows that the simple arc constructed by Newhouse and Peixoto can always be replaced by a stable one. For Morse – Smale diffeomorphisms defined on manifolds of any dimension, there are examples of systems that cannot be connected by a stable arc. In this connection, the question naturally arises of finding an invariant that uniquely determines the equivalence class of a Morse – Smale diffeomorphism with respect to the relation of connection by a stable arc (a component of a stable isotopic connection). In the article, the components of the stable isotopic connection of polar gradient-like diffeomorphisms on a two-dimensional torus are found under the assumption that all non-wandering points are fixed and have a positive orientation type.
We put forward a solution to the initial boundary value (IBV) problem for the nonlinear shallow water system in inclined channels of arbitrary cross section by means of the generalized Carrier–Greenspan hodograph transform (Rybkin et al. in J Fluid Mech, 748:416–432, 2014). Since the Carrier–Greenspan transform, while linearizing the shallow water system, seriously entangles the IBV in the hodograph plane, all previous solutions required some restrictive assumptions on the IBV conditions, e.g., zero initial velocity, smallness of boundary conditions. For arbitrary non-breaking initial conditions in the physical space, we present an explicit formula for equivalent IBV conditions in the hodograph plane, which can readily be treated by conventional methods. Our procedure, which we call the method of data projection, is based on the Taylor formula and allows us to reduce the transformed IBV data given on curves in the hodograph plane to the equivalent data on lines. Our method works equally well for any inclined bathymetry (not only plane beaches) and, moreover, is fully analytical for U-shaped bays. Numerical simulations show that our method is very robust and can be used to give express forecasting of tsunami wave inundation in narrow bays and fjords.
We analyze the main properties of soliton solutions to the generalized KdV equation u_t +[F (u)]_x+u_xxx = 0, where the leading term F (u) ∼ qu^α, α > 0, q ∈ R. The far field of such solitons may have three options. For q > 0 and α > 1 the analysis re-confirmed that all traveling solitons have ‘‘light’’ exponentially decaying tails and propagate to the right. If q < 0 and α < 1, the traveling solitons (compactons) have a compact support (and thus vanishing tails) and propagate to the left. For more complicated F (u) and α > 1 (e.g., the Gardner equation) standing algebraic solitons with ‘‘heavy’’ power-law tails may appear. If the leading term of F (u) is negative, the set of solutions may include wide or table-top solitons (similar to the solutions of the Gardner equation), including algebraic solitons and compactons with any of the three types of tails. The solutions usually have a single-hump structure but if F (u) represents a higher-order polynomial, the generalized KdV equation may support multi-humped pyramidal solitons.